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Name_________________________
Beam Deflection
Introduction
Engineers must look for better ways to build structures. Less material typically means that structures will be lighter and less expensive. Knowing the moment of inertia for different shapes is an important consideration for engineers as they strive to make designs lighter and less expensive.
Notes
Moment of Inertia (I): A measurement of the ________________ of an object, based
on its cross-sectional shape.
Measured in [in^4 ]
In general, a ___________ moment of inertia produces a
___________ resistance to deformation/bending.
What is the difference between beam A and
beam B?
Which beam will have a greater resistance
to bending if the same force is applied to
both beams? Why?
Calculating Moment of Inertia: ๐ฐ =
(b = base [in], h = height [in])
Calculate the moment of inertia for Beam A
geometry
I
1.5in 5 5in
12
I 20.8in
Stiffness
Greater
Greater
One is horizontal and one is vertical
A
Calculate the moment of inertia for Beam B
Beam A is _____ times stiffer than Beam B!
Modulus of Elasticity (E): A measurement of stiffness of an object, based on its
______________________________________.
Measured in [psi] (pounds per square inch)
In general, a ___________ modulus of elasticity produces a ____________
resistance to deformation.
Deflection (ฮ MAX ): A measurement of how much an object bends.
Factors that affect deflection
โ๐ด๐จ๐ฟ =
๐ญ๐ณ๐
๐๐๐ฌ๐ฐ
ex) Calculate the beam deflection for beam A.
I 5 5in^ 1.5in 12
I 1.54in
whats รtat^ inert
force applied^
t
length e
ฮ Max^
250
IBF 8.0ft
48 1,^
000
20.80in
ฮ Max^
Material properties
Greater Greater
Procedure
You will determine the weight of one of your classmates using nothing more than a standard 2x and a measuring device. This activity will provide you with a better understanding of Moment of Inertia and how it can be used to determine the strength of beams.
Preliminary lab calculations to determine beam Modulus of Elasticity
- Calculate beam Moment of Inertia
๏ฝ
3
xx
bh I 12
B โ width of the beam (in.)
h โ height of the beam (in.)
I โ Moment of Inertia (in.^4 )
Vertical Orientation Horizontal Orientation
IV = IH =
Position the beam as shown below.
- Measure the span between the supports. Record your measurement below.
Total Span (L) = __________in.
- Measure the distance between the floor and the bottom of the beam.
Pre-Loading Distance (D (^) PL ) = __________in.
(^1) 517in
3 482in
1.517 in 3 482 3.482in 1.517in
4875
- Position a volunteer (V 1 ) to stand carefully on the middle of the beam. Have a person on either side of the beam to help support the volunteer. Measure the distance between the floor and the bottom of the beam.
Applied Load Distance (D (^) AL ) = ___________in.
- Calculate the maximum beam deflection ( ๏^ MAX). ๏ (^) MAX = D (^) PL - DAL
๏ (^) MAX = __________ in.
6. Determine the weight of volunteer (V 1 ) using the classroom floor scale.
Volunteer weight (F) ____________ lb
- Calculate your beamโs Modulus of Elasticity (it is important to know that each beam will have its own specific Modulus of Elasticity) by rearranging the equation for beam maximum deflection to isolate (E). Show all work.
Rearrange the equation 3 MAX
FL ฮ = 48E I
to solve in terms of E
Substitute known values
Simplify and Solve
Note: An objectโs Modulus of Elasticity is a material-based property and stays the same regardless of orientation.
3 75
1 125
145
(^1 )
145 b^ 82.
48 E^ 1.^ n
48
EE
- Calculate volunteer (V 2 ) weight by rearranging the equation for maximum deflection to isolate (F). Show all work.
Rearrange the equation 3 MAX
FL ฮ = 48E I
to solve in terms of F
Substitute known values
Simplify and Solve
Determining Beam Deflection
- Using the information you collected and calculated in steps 1 โ 14, calculate the max deflection of the beam if volunteer (V 2 ) is positioned to stand on the beam in a vertical orientation.
3 MAX
FL ฮ = 48E I
Substitute known values
Simplify and Solve
- Verify your calculated max deflection answer and work to your instructor by having volunteer (V 2 ) carefully stand in the middle of the beam. Place a person on either side of the beam to help support the volunteer. Measure the distance between the floor and the bottom of the beam. Calculated deflection: _____________ Measured deflection: _____________ Instructor signature: ___________________________ Date: ________
F
48
EIA
Max
808 to
Eiiosint
48 1.800,000psi^ 1.103141 1
85in 3
8
5 L
1.45inapplied load^1 5.
waxbeam^ 0.15in
I
F
48EI Max
48
psi 5.34in^ 0.15in^ 125.51g
Practice Problem (to be completed INDIVIDUALLY)
- Complete the chart below by calculating the cross-sectional area, Moment of Inertia, and beam deflection, given a load of 250 lbf , a Modulus of Elasticity of 1,510,000 psi , and a span of 12 ft. Show all work in your engineering notebook.
Beam A B C D E F Common Name 2x6 2x6 2x8 2x8 2x10 2x Actual Dimensions (in.)
1.5 x 5.5 1.5 x 5.5 1.5 x 7.25 1.5 x 7.25 1.5 x 9.25 1.5 x 9.
Vertical or Horizontal Orientation
Cross- Sectional Area (in.^2 )
๐ด = 1. 5 ร 5. 5 = 8. 25 ๐๐^2
Moment of Inertia (in.^4 )
๐ผ =
- 5 ร 5. 53 12 = 20. 8 ๐๐^4 Beam Deflection (in.)
โ= โฏ = 0. 50 ๐๐
- Are your answers in the table above reasonable? How do you know? If they are not, discuss with a classmate and try them again on a fresh sheet of paper.
